There is another option: existence of a closed curve.
Usually this option can be analysed and excluded by using a nodal analysis, see [1]} for the case of unidirectional flows and [2]} for the case of constant vorticity. The aim of this paper is to study this option in different cases, when the water waves and supporting flow may contain stagnation points and critical layers and when the free surface is allowed to be overhanging, see [3]} and [4]} for discussions of problems arising in the nodal analysis.
